On Stationary Distributions for Dynamic Economies

By Edward PrescottHugo Hopenhayn
1990| Working Paper No. 1082

A problem that is arising with increasing frequency in dynamic economic analyses is the study of time invariant distributions. These arise in at least two classes of problems. The first is when the object of the research is an equilibrium distribution of agents indexed by some economic characteristics such as income, asset holding or employment status for individuals or the capital stocks of firms.’ The second is when the object of the research is the long run, behavior of a stationary stochastic process, as occurs in capital theory for the process induced by the optimal accumulation policy. Invariant distributions for such processes provide information on their long run behavior. In particular, the problem of uniqueness of an invariant distribution is closely related to the independence of this behavior from the initial data._x000B_ Existence arguments based on continuity conditions have been well studied.2 Recently, interesting economic models have been developed where nonconvexities or switching costs give rise to discontinuous stochastic behavior3, for which those arguments are not applicable. In some of these cases, however, the existence of stationary equilibria can still be established using different methods based on stochastic monotonicity conditions. In the first part of this paper we systematically develop this fixed point theory based on some recent results in probability._x000B_ Beyond the issue of existence of stationary distributions is the question whether the sequence of predictive probability distributions of future states has a limit and whether this limit is independent of the initial data. This has been the motivation of turnpike theory in stochastic growth models. The methods currently used have not proven easy, often